# Is there a difference between averaging individual regressions and including a random effect?

I have a bit of a theoretical question about random effects models and regression. If I have a set of clustered, longitudinal data (say repeated measurements of $$y$$ on a number of different individuals) is there really any difference, aside for adjustments to degrees of freedom, perhaps, to fitting a regression model for each individual and averaging the parameter estimates across models to get an overall average model (i.e. all fixed effects) vs. fitting a model with individual id as a random effect in which each of the individual's observations fall (i.e a mixed model)?

Thanks.

• Are you assuming you have an identical number of points per individual, or that you are doing a weighted averaging afterwards? – mkt - Reinstate Monica Jul 21 at 19:53
• Incidentally, your question could also be reframed as: "What's the difference between including individual as a fixed effect vs. a random effect?". And I discuss this a bit in my answer here: stats.stackexchange.com/a/289346/121522 – mkt - Reinstate Monica Jul 21 at 19:55
• @hi, mkt, I should have been more clear that I'm assuming equal sizes or weighted averaging. Thanks for the link. I'll give it a read now. – StatCurious Jul 21 at 20:42
• See Gelman's the Secret Weapon. – kjetil b halvorsen Nov 26 at 2:41